In the book there is this exercise:
Let E be an extension fiel of F, with $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over F but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$.
The suggested solution was:
It doesn't seem that they use that $\alpha$ is transcendental in F, or do they? Or is the statement valid without this assumption?